How do I define the domain of a function?

Question

I have a function of two variables, f[x_, y_], and I would like to restrict the domain to values of x and y greater than zero. How do I do this?

I also want to plot the function for restricted values. I have tried to use RegionFunction -> Function[{x, y}, ...], but this doesn't work.

Answer 1

You can also use Boole ... provided you are happy for your func to return 0 when the domain conditions are not met. For example:

f[x_, y_] := Boole[x > 0 && y > 0]  Cos[x] Sin[y] 

Plot3D[f[x, y], {x, -1, 2}, {y, -1, 2}]

Alternatively, you can set up a Piecewise function with explicit settings for what is to be returned (e.g. Indeterminate) when your conditions are not satisfied.

Answer 2

You may try "Putting constrains on patterns".

Mathematica provides a general mechanism for specifying constraints on patterns. All you need do is to put /; condition at the end of a pattern to signify that it applies only when the specified condition is True. You can read the operator /; as "slash-semi", "whenever" or "provided that".

Example:

Clear[f];
f[x_ /; x < 4] := x + 1;

or

f[x_] := x + 1 /; x < 4

Result:

(* defined *)
f[3]
4
(* undefined *)
f[5]
f[5]

Note:

In general, you can put /; condition at the end of any := definition or :> rule to tell Mathematica that the definition or rule applies only when the specified condition holds. Note that /; conditions should not usually be put at the end of = definitions or -> rules, since they will then be evaluated immediately.